A119771 Product of n^2 and n-th tetrahedral number: a(n) = n^3*(n+1)*(n+2)/6.

0, 1, 16, 90, 320, 875, 2016, 4116, 7680, 13365, 22000, 34606, 52416, 76895, 109760, 153000, 208896, 280041, 369360, 480130, 616000, 781011, 979616, 1216700, 1497600, 1828125, 2214576, 2663766, 3183040, 3780295, 4464000, 5243216, 6127616, 7127505, 8253840

Offset: 0

Author

Brandon Ang (xyz1236(AT)verizon.net), Jun 28 2006

Keywords

• easy
• nonn
• changed

Comments

If n is divisible by 10, then a(n) is divisible by 1000.

Examples

			a(25) = n^3*(n+1)*(n+2)/6 = 25^3*(25+1)*(25+2)/6 = 15625*26*27/6 = 15625*13*9 = 1828125.
		

Links

• Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Crossrefs

Cf. A000290, A000292, A061579.

Programs

Mathematica

a[n_] := n^3*(n+1)*(n+2)/6; Array[a, 35, 0] (* Amiram Eldar, Feb 13 2023 *)

Formula

From Alois P. Heinz, Feb 10 2023: (Start)

a(n) = Sum_{k=0..n} k^2 * A061579(n,k).

G.f.: x*(x+1)*(9*x+1)/(x-1)^6. (End)

From Amiram Eldar, Feb 13 2023: (Start)

Sum_{n>=1} 1/a(n) = 39/8 - 3*Pi^2/4 + 3*zeta(3).

Sum_{n>=1} (-1)^(n+1)/a(n) = 12*log(2) - 51/8 - 3*Pi^2/8 + 9*zeta(3)/4. (End)

E.g.f.: exp(x)*x*(6 + 42*x + 45*x^2 + 13*x^3 + x^4)/6. - Stefano Spezia, Aug 31 2025